Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $

TL;DR

This paper establishes isoperimetric inequalities with power weights in Euclidean space, showing that balls centered at the origin minimize weighted surface measure for fixed volume, with implications for inequalities and eigenvalue bounds.

Contribution

It introduces new isoperimetric inequalities with power weights, proving radial symmetry of optimizers and deriving sharp bounds for eigenvalues and inequalities.

Findings

Balls centered at the origin minimize weighted surface measure for fixed volume.

Results imply a weighted Polya-Sz"ego principle.

Establishes radiality of optimizers in Caffarelli-Kohn-Nirenberg inequalities.

Abstract

We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial \Omega } |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Sz\"ego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.